Water InfluxAdrian C Todd

Heriot-Watt UniversityINSTITUTE OF PETROLEUM ENGINEERING

Reservoir Performance Prediction Water InfluxLarge proportion of the worlds hydrocarbon reserves have an associated aquifer.These provide a major part of the energy for producing oil.Consideration that oil reservoir was originally occupied by water and oil has migrated in.Hydrocarbon and aquifer are therefore part of the same reservoir systems.Responding to the pressure changes resulting from oil production.

Water DriveWater drive most efficient displacing agent.Characteristics:Pressure decline is very gradualExcess water production in structurally low wells.Gas-oil ratio normally remains steadyA good oil recovery anticipated

Driving Force for Water DriveForce comes from response to pressure being lowered as a result of oil production.Aquifer is part of this system it also responds to the declining pressure.In MB fluid production due to compressibility of oil, rock & water.In aquifer the same is true.All the elements: expansion of hydrocarbon, water and rock as pressure declines.This compressibility gives understanding to water encroachment

Driving Force for Water DriveWater encroaches into oil reservoir in response to pressure reduction from well production.Pressure reduction comes from:Expansion of water due to pressure drop within aquifer.Expansion of hydrocarbons in the aquifer, if any.Expansion of rock, which decreases porosity.Artesian flow, if any, where outcrop is located structurally higher than HC accumulation, and water replenished at surface.

Driving Force for Water DriveAmount of expansion or fluid encroachment.The size of the aquiferPorosity and permeability of the aquifer rock.The presence of any artesian support.Amount of water flowing into HC reservoir depends on:Cross sectional area between oil reservoir and water zonePermeability of rock in aquifer.The viscosity of the water.

Aquifer predictionRequires considerable information of aquifer reservoir characteristics.The decline in pressure from oil and gas production moves with a finite velocity into the aquifer.This causes the aquifer, water and rock to expand.As long as the moving pressure disturbance has not reached the external limits of the aquifer, the aquifer will continue to provide expansion water to the HC reservoir.We refer to finite and infinite aquifers.Clearly no infinite aquifers, refers to time with respect to pressure disturbance reaching external limits.

Uncertainty of water drivethere are still more uncertainties attached to this subject in reservoir engineering, than to any other. This is simply because one seldom drills wells into an aquifer to generate reservoir characteristics. Instead these properties have frequently to be inferred from what has been observed in the reservoir. Even more uncertain is the geometry and areal continuity of the aquifer itself. The reservoir engineer should therefore consult both the production and exploration geologist. Due to these inherent uncertainties the aquifer fit obtained from history matching is seldom unique and the aquifer model may require frequent updating as more production and pressure data becomes available. Dake 1978.

Artesian vs. Compressibility.Artesian aquifers considered rare.Due to faulting, & pinch outs do not communicate to surface.Must be sufficient groundwater moving in to replace fluid withdrawl.Most water influx is considered due to expansion as result of pressure drop.

CompressibilityThe impact of a reduction in pressure is to cause expansion of the water and a reduction in pore volume.Difficult to separate the water expansion and rock compression.Usually combined to give effective water compressibility.Compressibility values are low 1.0x10-6 psi-1

Maximum Water InfluxAssuming no restrictions due to permeability maximum water influx associated with an aquifer, We, can be related to the volume of the aquifer, its compressibility and the pressure drop over itAquifer supported oil reservoirWhere:We water influxWi initial aquifer volumepi initial aquifer / reservoir pressurep current reservoir pressure, at original oil water contactc - total aquifer compressibilityThe main problem is determining the aquifer characteristic; geometry, size and flow characteristics.

Models for Water EncroachmentWater influx arises from pressure changes decompressing the aquiferIf the pressure can be determined then the volumetric changes can be obtainedPressure profiles generated as a result of decompression in oil and aquifer as a result of well bore production.

Diffusivity EquationThe flow rate at any radius r+dr is q.The rate of flow at radius r will be larger by the amount dq caused by:(I) expansion of the fluid q due to pressure drop dp over element dr.Expansion of (i) is too small and can be neglectedVolume of element is:

Change in volume dV due to pressure drop dp is;

Diffusivity EquationDarcys law for radial flowDifferentiating givesEquating equations for dq/drWhich giveswhereh is the diffusivity constant

Diffusivity EquationName comes from flow or diffusion of heatEquation applies to many conductive systemsIn radial flow for aquifer hydrocarbon system.The inner boundary is the extent of the hydrocarbon reservoir, the outer boundary is the limit of the aquiferIn flow within the oil reservoir The inner boundary is the radius of well bore and outer boundary the radius of the reservoir.

States of FlowDiffusivity equation show pressure is a function of time.As long as this exists dp/dt is not constant and flow is called unsteady state During the unsteady state period we need to analyse the pressure at each element across the radial symetry to determine the expansion.After a time dp/dt becomes constant and pseudo state state exists.All aquifers are finite, however there is a time period when a pressure disturbance has not reached the limit of the aquifer, during this time the aquifer behaves as infinite and unsteady state flow applies.After the boundary influences the behaviour pseudo steady state flow starts.

States of FlowDiffusivity equation indicates that state of flow is influenced by initial conditions and the boundaries which have a significant influence.Two boundary conditions must be specified:The inner boundary- the oil water interfaceThe outer boundary the limit of the aquiferConditions may be constant pressure, constant rate, closed boundary etc.Initial conditions at time =0, a uniform pressure distribution exists.To solve equation for water encroachment we need to specify the boundary and initial conditions

States of FlowIn water influx the common conditions are a closed system, no flow at the outer boundary.Constant rate or constant pressure at the inner boundary.In general constant pressure is used in aquifer modellingIn reservoir behaviour constant rate is assumed at the inner, well bore boundary

Schilthuis steady State ModelSimplest model.Aquifer very large so that pressure remains constant and it has high permeability, so no pressure gradient across aquifer.Hydraulic analogue:Aquifer tank pressure constant.Artesian aquifer or a very large aquifer

Schilthuis steady State ModelInitially aquifer and reservoir at same pressure.At intermediate pressure p, flow will be proportional to k & A of the pipe, Dp, pi-p, 1/m, and I/L.Maximum flow when p=0.If this rate greater than reservoir production, then rate of influx will equal rate of production and pressure will stabilise at a steady state value

This is analogue of Schilthuis steady state influx equation:

C is aquifer constant includes unchanging values of Darcys Law.In terms of rate:

Hurst Modified Steady StateThe analogue of this is that the tank is neither large nor replenished.The level in the tank falls and the potential of the aquifer falls.If this decline is exponential then is represented by:a is a time conversion constant

Van Everdingen and Hurst UnsteadystateThe equation will be developed later but is a model which is generally accepted in water influx modelling.The hydraulic analogue:

Van Everdingen and Hurst UnsteadystateSeries of tanks of increasing size.Initial all tanks at same level or pressure pi.As production occurs, pressure in reservoir tank causes water to flow from tank 1.Pressure in tank 1 causes flow from tank 2 and so on.Pressure drops are not uniform and will vary with time and production rate.They are progressive across the aquifer.

Pressure distribution for a constant rate of water influx

Pressure distribution for a constant boundary pressureIf there is an infinite number of tanks, the pressure will never stabilise

Analogue figure below represents cylindrical elements in an aquifer surrounding a circular reservoir.Analysis of the pressure in each element will enable expansion of water in each element can produce as a result of the effective compressibility from a pressure decline from pi to zero.

Unsteady state model Van Everdingen & HurstTank model indicates that unsteady state model is the exact solution.When influx is small Schilthuis steady sate can usually be used.For an active aquifer as pressure drop due to expansion moves out the expanding water has to move a greater distance to the oil or gas zone.The diffusivity equation provides the pressure, radius, time relationship.

Unsteady state model Van Everdingen & HurstExact analytical solution to this equation for a specific system will allow the calculation of rate of water influx.Van Everdingen & Hurst did this in 1949 for both the constant pressure case and the constant rate case.They produced a general solution based on dimensionless functions so that the solution is not specific but can be applied to different systems.

Unsteady state model Van Everdingen & HurstTo enable general application they produced a solution based on dimensionless functionsDimensionless time tD, and dimensionless radius rD .Dimensionless form of diffusivity equation is:where

Unsteady state model Van Everdingen & HursttD = time, dimensionlessT = time, secondsk = permeability, darcy.m = viscosity, centipoisef = fraction.c = effective compressibility aquifer, vol/vol//atmosro = reservoir radius, cm.re = aquifer radius

Converting to units; of time days, k millidarcies, m centipoise, c vol/vol/psi and r - feet

Unsteady state model Van Everdingen & HurstVan Everdingen & Hurst constant terminal rate boundary condition is used in well testing. They also derived the constant terminal pressure solution which is used in water influx calculations.This is the change in rate from zero to q due to a pressure drop Dp applied at the oil reservoir boundary at time =0More convenient to express the solution in terms of cumulative water influx into the oil reservoir rather than rate of influx.

Integrating the above equation therefore gives;

Unsteady state model Van Everdingen & HurstWhich givesWe = cumulative water influxdue to pressure drop Dp

Qt = dimensionless cumulative water influx functionThereforeIn oilfield unit termsWe - barrelsDp psiQt - dimensionlessorwhereB considered to be aquifer characteristic, terms do not change with time

Unsteady state model Van Everdingen & HurstVan Everdingen & Hurst have presented their solution in the form of dimensionless time ,tD and dimensionless water influx Qt.This enables their solution to be applied to any reservoir is radial in nature.Provided solutions for infinite and finite aquifers.Dake has provided graphical form of Van Everdingen & Hurst tables.

Qt vs. tD (Dake)Infinite aquifer

Qt vs. tD (Dake)

Qt vs. tD (Dake)

Qt vs. tD (Dake)Horizontal line indicates that the pressure drop impact has reached the limit of the aquifer and no further water is encroaching

Non circular systemsAlthough the solution is for full radial systems the solution can be applied to a non full radial system where a segment is considered.In this case

Summary of V&H expressions

Procedure for a fixed pressure dropDetermine aquifer characteristic BCalculate tDFrom tables or chart determine. Qt.Calculate We for fixed DpExercise 2

Application to a declining pressureThe V&H method to water influx is for a constant terminal pressure solution.The pressure at the reservoir aquifer contact is constant.In reality the reservoir contact pressure is declining continuously.

Application to a declining pressureV&H proposed a method by calculating the results for a series of fixed pressure drops and adding the solutions together.By superimposing the effects of a series of fixed pressure drops a steady declining pressure can be simulated

Application to a declining pressureWhat values do we use for Dp ?

What values do we use for Dp ?The gradual pressure drop is considered to be a series of fixed pressure drops.Need a method which will represent this.Shorter time steps will help to ensure that the pressure drops overlay the decline curve.

What values do we use for Dp ?First period Dp considered to be half of pressure drop(pi-p1).Second period Dp considered to be half of pressure drop period 1+half of Dp during second period.(pi-p1)+ (p1-p2)= (pi-p2)Third period Dp considered to be half of pressure drop period 2+half of Dp during third period.(p1-p2)+ (p2-p3)= (p1-p3)

History Matching Water InfluxA number of factors have a large influence on the pressure support from the aquifer.Size of aquiferGeometry of the aquiferMany of these factors are not available to the reservoir engineer.Only when production has started that one can determine the actual pressure support from various drive mechanisms.Can use the approach of Havlena & Odeh

History Matching Water InfluxWater influx = Initial water volume x pressure drop x aquifer compressibility.This ignores unsteady state behaviour.Need to consider the V&H approach

Water Drive. No Original Gas Capre/ro too small or too largeGeometry not correctOnce values give good match based on previous production can be used to predict future performance

Used in Piper field (UK) to determine aquifer sizeMatch for infinite aquifer

Very Small AquifersSmall aquifers can assume steady state flow and pressure drop quickly transmitted to the aquifer boundaryA straight line should give a slope of B and intercept N.

Water Drive Gas Cap Known SizeSimilar to beforeThis is a straight line if geometry of aquifer and time are correct

Small aquifer Gas Cap Known Size

Reservoir Performance Using Unsteady State Equation and the MB equationWith both water influx and MB equation there are two unknowns, We & Pressure.It is not straightforward to predict performance. It is a trail and error approach.1. Collect all available reservoir & subsurface data.2. From past production data we can calculate the aquifer constant B. MB gives We and for a known aquifer can determine B.Important to do this for a number of past times.

Reservoir Performance Using Unsteady State Equation and the MB equationUsing unsteady state equation and knowing time and pressure drops can determine SDpQt.Value of aquifer characteristic, B.

Reservoir Performance Using Unsteady State Equation and the MB equationUsing past trends of production, oil, gas & water, the future trends are projected.Trail & error approach.1. Estimate pressure after say 6 months2. Gross We calculated by MB and USS equations3. If both agree pressure assumed is correct. If not another selected until agreement reached.Procedure carried out for each time period.Different combinations of production rates should be used and associated decline predictions of pressure made against each set of production values.

Other approaches to We predictionV&A USS method although giving a good prediction it is tedious in its application.Fetkovitch in 1971 provided an approach based on the productivity approach.It can only be applied however to finite aquifers.In 1960 Carter & Tracy produced a method based on the constant terminal rate solution which does not require a superposition of values to apply to a declining pressure as does the V&A USS method.